Against Tabula Rasa, Part 3: Degrees of Belief

In my first post, I argued that the judge is constrained not by possibility, but by rationality. In my second post, I argued that the judge’s starting points for a debate bear on in-round disputes because of conflicting assumptions. My next argument comes from another aspect of rationality. In this post, I set up the general framework for the argument, but the argument itself will be in the next post.
(Note: In what follows, I may be making simple mistakes or misunderstandings. If you know a thing or two about the technical stuff below, then please correct me in a comment not only if you think I’m wrong about something important, but also if I misuse any terms or concepts.)

We don’t just believe or disbelieve propositions. We have greater or lesser degrees of belief in propositions. These degrees of belief are called credences. For example, I have greater credence in the proposition “2 + 2 = 4” than I have in “It will rain in Oxford tomorrow,” although I believe both.

We can account for degrees of belief through Bayesian statistics. Here are the technical parts:

We have degrees of belief in all well-defined propositions. We can call your initial credences your priors. For example, I just rolled a six-sided die. Before looking at it, my credence in the proposition that I rolled a four is 1/6. When you get new evidence, you update your credences into posterior degrees of belief. Your posterior credences are your degrees of belief after accounting for your evidence. So, after I observe that I rolled a four, my posterior credence in the proposition that I rolled a four is close to 1 (but not exactly 1, since my eyesight may be misleading, or something else may be going wrong). The updating process follows a rule of inference that we can ignore for now.

I understand tabula rasa to be the following view:

The judge’s priors should be as neutral, or uninformative, as possible. When a debater wins an argument for some claim, the judge’s credence in that claim increases. A perfect judge would update her credences based on the arguments in the debate as an ideal Bayesian reasoner. The judge’s posterior credence in the resolution (or in who did the better debating, who she should vote for, or whatever other question you think should determine the ballot) determines her decision. A judge’s decision is better or worse to the extent that it approximates that ideal.

I realize that no proponents of tabula rasa have actually stated their view in this way. But, to be honest, I don’t think they have stated their view precisely at all. I think the above is the best version of what they’re after. (They can’t be after “no priors allowed,” since that’s both impossible and irrational. And, as I argued in the last post, the judge needs priors to evaluate disagreements that appeal to conflicting assumptions.)

But how exactly is the judge supposed to have maximally uninformative priors? In the next post, I argue that this question raises some difficult problems for the tabula rasa view.

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The way we instruct judges on the concept of tabula rasa is like this: if the affirmative team is arguing that the sky is green and the negative team does not provide refutation that the sky is actually blue, then you must accept the argument that the sky is green even though you know from personal experience that the sky is blue.

AndrewG

Describing the tabula rasa view through the lens of Bayesian Inference seems convoluted. Just look at the definition you gave – it's not particularly informative. The fundamental problem in applying this framework, is that the concept of 'winning an argument' is very difficult to divorce from one's beliefs (both prior and posterior) about what is being argued.Entertaining this framework though, I would say that in the abstract, a tabula rasa judge is one whose posterior beliefs are minimally impacted by one's prior beliefs – and maximally impacted by 'won' arguments. Let's take the case where someone argues that the Sky is Green (SG), and the argument is Dropped (D). What we want to model is the idea that the judge's belief that the Sky is Green (SG) is very high given the argument is Dropped (D) – because a tabula rasa judge must accept dropped arguments. So, using Bayes' Theorem, we have the following:P(SG|D) = P(D|SG)*P(SG)/P(D)Here P(SG) and P(D) are our prior beliefs that the Sky is Green, and that such an argument would be Dropped respectively. P(D|SG) is the likelihood that such an argument would be Dropped given that the Sky is Green. For P(SG|D) to be high (essentially 100%), one needs P(D|SG)*P(SG) to be high, and/or for P(D) to be low. In other words, the judge must believe that it is both likely that the argument would be Dropped if the Sky was actually Green, and must believe that it is at least possible for the Sky to be Green. As well, the judge must believe that it is unlikely for the argument to be Dropped, irrespective of whether or not it's true. Interpreting this further, the tabula rasa judge believes some combination of the following (with many different potential weightings):1) An argument is likely to be dropped if its conclusion is true. P(D|SG) is high.2) Any conclusion is at least possible. P(SG) is at least > 0.3) Arguments are unlikely to be dropped. P(D) is low.Hopefully this is a bit more clear. Your original explanation intended to explain that the tabula rasa judge must have P('Any argument') (in this case SG) = 0.5 (neutral). I don't think this is quite correct. Instead, I think the judge does have prior beliefs (i.e. P(Arg 1) != P(Arg 2) necessarily – as long as P(Arg N) > 0), but the in-round interplay completely dominates the tab judge's prior beliefs. As such, (1) and (3) are the dominant effects. Of course, both (1) and (3) are silly, and this is the fundamental reason why tabula rasa judging is odd.Now, the difficulty in applying the model comes in when, instead of using 'Argument is Dropped' (D), we use 'Argument is Won' (W). For, the concept of 'winning' an argument, except in the case where the argument is dropped, is quite nebulous. (This is one reason why tabula rasa judges tend to default to voting for dropped arguments.) I would be interested to see how you think this model could be applied to arguments where there has been clash.

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